Timeline for Existence of non-trivial solution to non linear polynomial system
Current License: CC BY-SA 3.0
5 events
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| Jun 16, 2013 at 19:58 | comment | added | user34985 | Thank, just a few last clarification: 1.Regarding the first question, what about over $\mathbb{R}$, which is my case? what are the restrictions? what's are the points that it should be take care? 2.Regarding question 5- In fact I need for other problem, the general case of two quadratics forms. I tried to look on that most general case, but it seems that the equations derived from Groebner bases are hard, so if there some source which give a closed solution for that case, it would be great. Nonetheless i'm always happy to learn new stuff, and surely I will read yours lecture notes. | |
| S Jun 16, 2013 at 18:09 | comment | added | user34985 | 5. Is there a way to utilize the fact that my equations are quadratic forms? I know that for two equations with two variables, the Sylvester matrix give an immediate solution. I also found that there are known result for three quadratic equations with three variable (using resultant). Isn't any general result for any two homogeneous equations with three variable?? Thanks a lot, | |
| S Jun 16, 2013 at 18:09 | comment | added | user34985 | 3. If I choose an ordering (let say $x>y>z$) and get a condition, let's say about $x$ (as in the example). Is it possible to substitute it in the original equations, and get a new Groebner bases using different order of the variable (for exmple $z>x>y$)? Can I solve directly the original equations without using Groebner bases (after substitute $x$)? 4. Is there any recommended book about Groebner bases? I am not a mathematician so I don't familiar with all the terminology such as Ideal? A book with full examples solution will be the best one for me. | |
| S Jun 16, 2013 at 18:09 | comment | added | user34985 | thank you very much!! I am not familiar with the Groebner bases so I have a few more questions: 1. As far as I understood, Groebner bases give me a new set of equations, which theirs solution is exactly as the original equations. This is always true? is there any restrictions on the coefficient? 2. I see that (using computer software such as 'mathematica') for each ordering of the variable i get different bases. Can I just choose any order which will give me easiest set of equation? | |
| Jun 15, 2013 at 21:46 | history | answered | Dietrich Burde | CC BY-SA 3.0 |